Dynamic frequent subgraph mining algorithms over evolving graphs: a survey

Scope and motivation of the survey

When we scan the literature related with algorithms on dynamic subgraph graph mining over dynamic graphs we come across few surveys (Fournier et al., 2020; Chaudhary & Thakur, 2018). They concentrate on discovering frequent subgraphs, evolution rules, motifs, subgraph sequences, recurrent and triggering patterns, and trend sequences on graphs where dynamic aspect can be incremental based, snapshot-based or interval based. Our motivation on the other hand is to be able present a comprehensive and focused literature review on “dynamic frequent subgraph mining algorithms” only. Another distinction of our survey from the previous surveys we consider only “evolving graphs” with increments either as a series of small graphs or stream of nodes and edges.

We try to present and compare frequent subgraph mining algorithms over evolving graphs. We analyze them under two main groups where the first group is exact algorithms whose objective is to find all frequent or closed subgraphs of the evolving graph and the second groups is approximate algorithms whose objective is to discover frequent subgraphs by employing a sampling strategy.

In summary, this article makes following contributions:

•   It gives detailed background information related with graph basics, increment types of dynamic graphs, frequent subgraph mining, dynamic frequent subgraph mining and sampling strategies in the context of approximate dynamic frequent subgraph mining.

•  It gives a qualitative literature survey on dynamic frequent subgraph mining algorithms designed for evolving graphs. Review contains a comparison of these algorithms with respect to their various characteristics. In addition, each of them is introduced briefly.

•  It gives another qualitative literature survey on approximate dynamic subgraph mining algorithms again with respect to their various characteristics. This time the characteristics include the ones related with their sampling strategy as well. Similar to the previous part, the review contains a comparison of the algorithms in addition to the brief introduction of each.

•  It tries to contribute both as an introduction and as a chronological guide to recent advances and opportunities in this specific research area of frequent subgraph mining over evolving graphs.

Outline of the survey

The rest of this survey flows as follows: In “Survey Methodology” section, we address our research questions, data sources and research strategy, inclusion/exclusion criteria. In the “Background” section, we introduce the preliminary information required in explaining and comparing the algorithms of the survey. In the “Frequent Subgraph Mining Algorithms for Evolving Graphs” section, we evaluate dynamic frequent subgraph mining algorithms within the scope of the survey with a detailed comparison and discussion. In “Approximate Frequent Subgraph Mining Algorithms for Evolving Graphs” section we outline approximate frequent subgraph mining algorithms again with a comparative perspective as well. In “Research Opportunities” section, we try to highlight potential future research directions in the field. Finally, in “Conclusion” section, we summarize all our findings in the survey.

Portions of this text were previously published as part of a thesis (https://openaccess.iyte.edu.tr/bitstream/11147/12616/1/10147615.pdf)’.

Data sources and research strategy

We utilized Google Scholar, ACM Digital Library, IEEE Explore, Wos, Arxiv, Siam Journal and Springer Link in finding related literature. The search strings that we used in the search are “frequent subgraph mining” AND “evolving graph” AND “approximate subgraph discovery” AND “frequent patterns” AND “dynamic networks” in all the search engines. Table 1 presents all the data sources, links and the search strings used in data collection. Same in the second column means that the search strings are the same as indicated in the first row of the table.

Criteria for inclusion/exclusion

After searching all related literature from different search engines, we include only (i) research articles written in English, (ii) research articles published between the years 2006–2024, and (iii) research articles received from peer-reviewed resources. We exclude (i) parallel algorithms of the field and (ii) algorithms focusing on complex graphs such as attributed graphs.

No parallel/only frequent subgraphs/no complex graph types/incremental not snapshot based not window based.

Graph basics

A graph is defined as a set of vertices (nodes) that are interconnected by a set of edges. Example of a graph is shown in Fig. 1. A graph G is an ordered pair (V, E) consisting of a set of vertices V = {v1, v2, v3, v4, v5} and V is connected to each other by and a set of edges E = {e1, e2, e3, e4}. A label function, L, maps a vertex or an edge to a label.

Assume subgraph G ’(V’, E’) is a subgraph of the graph G (V, E), where edges and vertices are subsets of E and V respectively. Figure 2 shows examples of subgraphs such that S1, S2, and S3 are subgraphs of G of Fig. 1.

Subgraph isomorphism: Given two undirected graphs G and H. There is an isomorphism between G and H, if there is a bijection f between their vertices (f: V(G) →V(H)), Hence, two vertices u, v are adjacent to each other in G if and only if f (u), f (v) are adjacent in H. These two graphs are topologically identical. In Fig. 3, the subgraph g₁ isisomorphic to the subgraph (u₃, u₄, u₅), where (u₃, u₄, u₅) is a subgraph of the input graph G. This is also called an embedding g₁ in graph in G.

Static graph: A graph G = (V, E) consists of a set of nodes V, and a set of edges E ⊆ V ×V, where V and E do not change over time.

Dynamic graph (evolving graph): An evolving graph G_D = (V_D, E_D) consists of a set of nodes V_D, and a set of edges E_D ⊆ V_D × V_D. G_D is changed by node additions or deletions, edge additions or deletions over time. Figure 3 shows an example of a dynamic graph at three points of time (t₁, t₂ and t₃). At time t₂, there is a deletion of the edge u₇₋u₈. At time t₃, edge u₇₋u₈ andedge u₃₋u₁₀are added.

Increments for dynamic graphs

In the context of dynamic graphs, graph increments can be represented as (a) a series of small graphs, or (b) a stream of node and edge updates (Ray, Holder & Choudhury, 2014).

Increments as a stream of nodes and edges: In this kind of dynamic graphs, the increments consist of nodes and edges that change over time, and it can be addition or deletion.

Dynamic graph (evolving graph): An evolving graph G_D = (V_D, E_D) consists of a set of nodes V_D, and a set of edges E_D ⊆ V_D × V_D. G_D is changed by node additions or deletions, edge additions or deletions over time. Figure 3 shows an example of an edge updates, it shows the dynamic graph at different time points t₁, t₂ and t₃ when edges are coming as updates. At time t₂, there is a deletion of the edge u₇-u_8. At time t₃, edge u₇-u₈ and edge u₃-u₁₀ are added.

Increments as a series of small graphs: In this kind of dynamic graphs, the increments are done by using a series of graph objects, i.e., each object in the stream is considered as a (static) graph snapshot.

Series of small graphs: Given a sequence G_ts of n graphs {G₁, . . . , G_n} with G_i = (V_i, E_i) for 1 ≤ i ≤ n. We define G_ts to be a time series of graphs if V₁ = V_i for all 1 ≤ i ≤ n. G_i is the i-th state of G_ts (Borgwardt, Kriegel & Wackersreuther, 2006).

Figure 4 shows an example of a dynamic graph when the input is a series of small graphs. It shows a graph input, where each incoming object is an entire graph.

Frequent subgraph mining

The objective of a frequent subgraph mining (FSM) task is to find all frequent subgraphs in a given graph or a set of graphs that occur more than a given user-defined threshold (Aggarwal & Wang, 2010). Number of occurrences of a subgraph (g) in a graph dataset is called support of subgraph (g). If the support of subgraph is more than minimum support threshold that is defined by the user, then this subgraph is a frequent subgraph (Dinari & Naderi, 2014).

Figure 5 illustrates an example of finding frequent subgraphs. Input is a database of graph transactions, undirected simple graph (no loops, no multiples edges), each graph transaction has labels associated with its edges and vertices, transactions might not be connected and a minimum support threshold σ; example (60%). The output is frequent subgraphs that satisfy the minimum support threshold, and each frequent subgraph is connected.

Frequent subgraph mining task is computationally challenging since number of possible pattern experiences a combinatorial explosion with the maximum number of vertices and vertex labels. Finding isomorphic subgraphs is the other challenge of this process.

Dynamic frequent subgraph mining

It is the process of finding all frequent subgraphs on evolving graph. In the example presented in Fig. 3 given an input dynamic graph G and support threshold 2, let us see the status of subgraphs g₁ and g_2. Addition of an edge to input graph increases the support of one or more subgraphs, removal of an edge of the input graph decreases the support of one or more subgraphs (Abdelhamid et al., 2017). At time t₁, the subgraph g₁ has two matches in G, however the subgraph g₂ has only one match. As a result, g₁ is frequent subgraph and g₂ is not frequent subgraph. At time t₂, there is a deletion of the edge u₇₋u_8, so the number of embeddings of g₁ decreases to one, however the number of embedding of g₂ does not change. Therefore, both subgraphs g₁ and g₂ are not frequent. At time t₃, there is an addition of edge u₃₋u₁₀, this addition increases the number of matches of g₂ to two; so, g₂ becomes frequent.

Graph sampling

Graph sampling is needed in network analysis, to keep the graph scale small while capturing the properties of the original graph, graph sampling provides an efficient, yet inexpensive solution for network analysis (Wang et al., 2011). Another reason for using graph sampling; it is sometimes only the solution since obtaining all data for a graph is not permitted or is very time-consuming. Thus, we must obtain the properties of the graph by sampling.

Graph sampling can make the graph size smaller while keeping the characteristics of the original graph (Sahu et al., 2023; Wang et al., 2011). Graph sample of Graph G = (V, E) is defined as G_s = (V_s, E_s) where V_s ∈ V, E_s ∈ E and E_s ⊆ {(u, v) —u ∈ V_s, and v ∈ V_s}. There are different sampling methods like Node Sampling, Edge Sampling, Traversal-based sampling (Zhang et al., 2017) and Sampling with Neighborhood (VSN) (Hu & Lau, 2013). Our intention is not give detail review of graph sampling strategies and methods but we would like to mention the ones used in the context of frequent subgraph discovery by static and dynamic frequent subgraph mining algorithms.

Approximate dynamic subgraph mining

FSM task on large graphs is computationally challenging due to combinatorial explosion in possible number of vertex and vertex labels of subgraphs and subgraph isomorphism operations needed. This process is generally NP-complete and exact algorithms tend to scale poorly. Using a sample of the graph whenever the approximate results serve the purpose is a solution. Another reason for using sampling is for the cases when the entire graph cannot be retrieved. Main challenge of sampling is to provide theoretical guarantees on the quality of the sample. We see a good example sampling based randomized algorithm MaNIACS (Preti, De Francisci Morales & Riondato, 2023) for computing high quality approximations of the collection of the frequent subgraphs in a vertex-labeled graph. The upper bound of the estimation corresponding to maximum allowed estimation error relies on empirical Vapnik–Chervonenkis dimension (Riondato & Upfal, 2018). MaNIACs algorithm is designed for static graph and used for creating sample graph that is used in discovering approximate frequent subgraphs.

When we deal with approximate solutions in dynamic frequent subgraph mining we see four sampling algorithms Simple Random Sampling (Yates, Moore & Starnes, 2002), Neighborhood Sampling (Ray, Holder & Choudhury, 2014), Reservoir Sampling (Vitter, 1985) and Sampling using Vapnik–Chervonenkis dimension (Nasir et al., 2021).

Simple random sampling: Yates, Moore & Starnes (2002): In this sampling scheme an item from the data stream is picked or not picked for the sample with a user defined probability. The inclusion probability can be given to the algorithm as an input parameter. For example if inclusion parameter is 0.5, then each item has an equal probability of being picked or not. It is a simple scheme but we can see two disadvantages: (i) size of the sample grows as the size of the input data stream grows and (ii) quality of the sample can only be guaranteed when the sample size exceeds specific size (Anis & Nasir, 2018).

Neighborhood sampling: Ray, Holder & Choudhury (2014): We see this sampling approach in StreamFSM algorithm, which reports the frequent subgraphs that are estimated to be found in the entire graph as each batch, arrives. When a new edge arrives to the graph, a number of nodes that are one hop away from this edge point is selected. Then edges between these nodes are included in the sample. The idea is to restrict the changes to the set of frequent subgraphs to the locality of this new edge. Selecting the nodes, however, is dependent on the domain. Domains that have a high frequency of large star-shaped patterns, where the edges in the star have similar labels, require some pruning. In other domains to select all the nodes in the immediate neighborhood of the edge is possible. These extracted neighborhoods now become graph transactions, which are input to a graph transaction miner like gSpan (Yan & Han, 2003). Selecting a higher number of neighbors increases the running time and the accuracy, while selecting a lower number of neighbors does the reverse.

Reservoir sampling: Vitter (1985): It is a fixed-size randomized sampling scheme. It maintains a fixed-size uniform sample of the data stream as first introduced (Gemulla, Lehner & Haas, 2006). The size of the sample is provided as the input parameter. The algorithm initializes with a fixed-size sample. When the sample reaches to the size defined as input, the randomly chosen item from the data stream is added to the sample and an existing item from the sample is deleted. Therefore, the reservoir-of sample size is kept constant. This random pairing is a fully dynamic algorithm for reservoir sampling. It maintains additions and deletions to and from the sample (Anis & Nasir, 2018).

Sampling using Vapnik–Chervonenkis dimension: TipTap algorithms Nasir et al. (2021): try to maintain a uniform sample of connected k-vertex subgraphs with an neighborhood exploration procedure. The sample size is not fixed like rs, it is flexible and allows desired approximation quality. In defining sample complexity bounds, Vapnik–Chervonenkis dimension (Riondato & Upfal, 2018), a key concept from statistical learning theory is used and it helps to derive sufficient sample size as in the MaNIACS algorithm (Preti, De Francisci Morales & Riondato, 2023).

In this section, we started by giving some graph basics like the definition of a graph, subgraph, subgraph isomorphism, graph embedding, static graph, dynamic graph. In addition to basic graph definitions, increments as series of small graphs or as stream of nodes and edges on evolving graphs are explained each with an example. Then the subject processes of the survey like frequent subgraph mining, dynamic subgraph mining, graph sampling and approximate dynamic frequent subgraph mining are introduced. All of these are required background information to understand the algorithms surveyed and compared in the following sections of the article.

Frequent subgraph mining algorithms for evolving graphs

In this section, we introduce the literature on frequent subgraph mining algorithms for dynamic graphs. We start with presenting a detailed comparison table where all related research is given with their general input/output attributes (input graph type, increment type, increment content and output type) and programming attributes (graph representation, data structure, algorithmic approach and base algorithm). We then discuss each algorithm introduced in the table. We wrap up the section with a discussion of subject algorithms.

Although the number of algorithms are not as many as the static counterparts, in dynamic frequent graph mining literature, various algorithms have also been proposed using different approaches and characteristics. Table 2 gives a comparison among these algorithms with their attributes. The first group of attributes in the table are related with the input and output of the algorithms whereas the second group of attributes are related with the internal characteristics of the algorithms. The increments type, as the listed algorithms are incremental algorithms, the increments can be a series of small graphs or a stream of nodes and edges. The graph type of the input graphs can have one of the following possible types: Undirected labelled graphs (UL), undirected graphs (U), directed graphs (D), various kinds of graph data (V) or labeled graph (L). The increments can have Addition (A) or Deletion (D) of Nodes/Edges or Addition (A) of Edges. Output type indicates the essential purpose of each algorithm; it can be to find all frequent subgraphs or closed subgraphs.

The second group of attributes starts with the graph representation, which is considered as one of the most effective attributes on the consumption of runtime and memory and can be adjacency matrix, adjacency list and canonical labelling. The data structure represents data structure that is used in the algorithm and it can be DFS tree, Suffix trees, DS-Tree, DS-Table, DS-Matrix or Index Structure. The algorithmic approach shows the frequent subgraph mining approach of the algorithm where the possible values can be Apriori or Pattern growth. Programming approach shows whether the algorithm utilize parallel programming or not. Base algorithm gives information related with static base algorithm since some incremental algorithms are developed based on extensions of the static ones.

The Dynamic GREW (Borgwardt, Kriegel & Wackersreuther, 2006) investigates how pattern mining on static graphs can be extended to time series of graphs. Specifically, it handles dynamic graphs with edge insertions and edge deletions over time. They define a frequency and provide algorithmic solutions for finding frequent dynamic subgraph patterns. Existing subgraph mining algorithms can be easily integrated into this framework to make them handle dynamic graphs. Experimental results in the paper on real-world data confirm the practical feasibility of proposed approach, and what the limitations of Dynamic GREW are; it assumes that the input dynamic graph has a fixed set of nodes, and edges are inserted and deleted over time. Also, there is an extra overhead to identify dynamic patterns. It misses some interesting patterns.

Berlingerio & Bonchi (2009) introduced Graph Evolution Rule Miner (GERM), a novel type of frequency based pattern that describe the evolution of large networks over time, at a local level. The input for this approach is a sequence of snapshots of an evolving graph; the main purpose is to mine the rules that describe the local changes in it. This approach uses the support based on minimum image to extract patterns which frequency is greater than a minimum support threshold. After that, graph-evolution rules are extracted from frequent patterns that satisfy a given minimum confidence constraint, the rules extraction framework is similar to it in classical rule mining. Experiments are done on four large real-world networks (two social networks, and two co-authorship networks), using different time granularities. Experiments approve feasibility and the utility of o framework. The limitations of GERM: it is designed for undirected graphs, nodes and edges are only added and never deleted. It assumed that node and edge labels do not change over time.

The algorithm in Miyoshi, Ozaki & Ohkawa (2011) handles the problem of mining frequent patterns and valid rules representing graph evolutions (structural changes) in a graph with time information. They propose an efficient technique for extracting representative patterns and rules; they use graph-based summarization of discovered rules. This done by using certain measures provided by the summary, so it is expected to find more interesting information which are difficult to be discovered by the traditional support and confidence measures. Proposed algorithm based on gSpan and Germ, it differs from gSpan that it handles single graph input and work on graph patterns have time points, it differs from Germ that proposed method that handles multi edges.

In Bifet & Gavaldà (2011), the first work on close stream while only two frequent close graph algorithms on static graphs are introduced. Bifet et al. proposed new method for mining frequent closed subgraphs. The method is IncGraphMiner, it works on frequent weighted closed graph mining. this method works on coresets of closed subgraphs, compressed representations of graph sets, and maintain these sets in a batch-incremental manner; it handles the potential concept drift. The proposed algorithm is based on close graph, which takes time overhead to summarize the patterns.

In Bifet & Gavaldà (2011), the first work on close stream, they proposed frameworks are for studying graph pattern mining on time-varying streams. Two new methods for mining frequent closed subgraphs are presented. The methods are WinGraphMiner and AdaGraphMiner, which work on frequent weighted closed graph mining. All methods work on corsets of closed subgraphs, compressed representations of graph sets, and maintain these sets in a batch-incremental manner but use different approaches to address potential concept drift. The above three algorithms are based on close graph which takes time overhead to summarize the patterns.

Most algorithms are programmed in serial approach while very few are developed in parallel manner like span (Lakshmi & Meyyappan, 2013). The span (Lakshmi & Meyyappan, 2013) is based on gSpan. It handles special class of undirected labelled simple graphs, graphs with unique or no labels. Parallel programming is used in order to decrease the complexity. Proposed solution, is applicable in cases where graph dataset fits in main memory. Two techniques, DFS lexicographic order and minimum DFS code are introduced in gSpan. With the help of these techniques, a novel canonical labeling system is constructed which supports DFS search. However, finding minimum DFS code is still an NP- complete problem. That is why a modified DFS representation is proposed. This representation utilizes advantages of gSpan, and allows parallel programming on multi-core processing technology to improve the performance. The number of duplicate graphs generated are slightly more than gSpan algorithm since sub graph mining from frequent single edge graphs are done in parallel.

The StreamFSM Ray, Holder & Choudhury (2014) discovers the frequent subgraphs in a graph, represented by a stream of labeled nodes and edges. In this model, updates to the graph arrive in the form of batches that contain new nodes and edges. The proposed method continuously reports the frequent subgraphs that are estimated to be found in the entire graph as each batch arrives. It is evaluated using five large dynamic graph datasets: the Hetrec 2011 challenge data, Twitter, DBLP and two synthetic datasets. It is evaluated against two popular static large graph miners, i.e., SUBDUE and GERM. Experimental results show that it can find the same frequent subgraphs as a non-incremental approach applied to snapshot graphs, and in less time. The drawback of the StreamFSM algorithm: In terms of several parameters that have to be tuned in order to get the optimal performance in terms of time and accuracy/interestingness of results. Also, it assumes that we have access to the entire graph as the graph grows. This assumption will not work in a real world-streaming scenario (Ray, Holder & Choudhury, 2014). In addition, it only handles increments with additions.

The IncGM+ Abdelhamid et al. (2017) is a fast incremental approach for continuous frequent subgraph mining on a single large evolving graph. It adapts the notion of “fringe” to the graph context, which is the set of subgraphs that are on the border between frequent and infrequent subgraphs. IncGM+ maintains fringe subgraphs and utilize them in the search space pruning. In order to increase the efficiency, an efficient index structure is proposed to maintain selected embeddings, with minimal memory overhead. These embeddings are employed to avoid subgraph isomorphism operations. Furthermore, the proposed system supports batch updates. Experiments are done using large real-world graphs, it verifies that IncGM+ outperforms existing algorithms by up to three orders of magnitude, scales to much larger graphs, and consumes less memory. The limitation of IncGM+ that it still needs to enumerate and track an exponential number of candidate subgraphs.

DyFSM Chen et al. (2023) is a frequent subgraph mining algorithm designed for dynamic databases where incremental updates and decremental updates are handled. It uses a buffer named Fringe- subgraph collection to keep these updates. A novel representation based on DFS code used by DyFSM allows enumeration of frequent subgraphs rapidly without subgraph isomorphism computations.

Approximate frequent subgraph mining algorithms for evolving graphs

In this section, we will start by talking on the challenges of approximate dynamic subgraph mining. Then, like the previous section, we will introduce the literature on approximate frequent subgraph mining algorithms for dynamic graphs. We start with presenting a detailed comparison table where all related research is given with their general input/output attributes (input graph type, increment type and increment content), programming attributes (graph representation, data structure and objective) and sampling attributes (sampling strategy, statistical proof and data in the sample). We then brief each algorithm introduced in the table. We wrap up the section with a general summary of the facts of subject algorithms.

FSM task on large graphs is computationally challenging due to combinatorial explosion in possible number of vertex and vertex labels of subgraphs and subgraph isomorphism operations needed. This process is generally NP-complete and exact algorithms tend to scale poorly. Using sample of the graph whenever the approximate results serve the purpose is a solution. Another reason for using sampling is for the cases when the entire graph cannot be retrieved. Main challenge of sampling is to provide theoretical guarantees on the quality of the sample. In fact we see examples sampling based FSM algorithms in the context of static graphs like MaNIACS (Preti, De Francisci Morales & Riondato, 2023), Ap-FSM (Bhatia & Rani, 2018; Purohit, Choudhury & Holder, 2017; Zou & Holder, 2010) with or without sample quality guarantees.

We already introduced in background section, sampling methods used in the context of approximate dynamic frequent subgraph mining. In the remaining part of this section, we try to analyze sampling based FSM algorithms on evolving graphs. These approximate algorithms face challenge of maintaining good quality sample in addition to handling increments done to the graph and complexity of base FSM algorithms.

As already mentioned, approximate dynamic frequent subgraph mining algorithms work on the dynamic sample instead of dynamic graph and their results are approximate. They require a sampling method in order to keep dynamic and high quality sample of input graphs. Table 3 gives a comparison among these algorithms with their attributes. The first group of attributes in the table are related with general input and output of the algorithms whereas the second group of attributes are related with the internal characteristics of the algorithms and third group of attributes denote the characteristics of the sampling scheme. The increments type, as the listed algorithms are incremental algorithms, the increments can be a series of small graphs or a stream of nodes and edges. The graph type of the input graphs can have one of the following possible types: various kinds of graph data (V), connected undirected graphs (CU) and (L) labeled. The increments can have nodes/edges where each column can have, Addition (A) or Deletion (D).

The second group of attributes starts with the graph representation, which is considered as one of the most effective attributes on the consumption of runtime and memory and can be DFS Tree or Array with hashed based index or Array with hash map. The data structure represents data structure that is used in the algorithm and it can be dictionary data structure or Canonical label. The objective of this group of attributes is to minimize execution time or maximize accuracy. The third group of attributes are related with sampling scheme of the algorithm where sampling strategy gives information if sample size is fixed or flexible. Statistical proof shows whether the algorithm uses an upper bound formula and unbiased estimator. In addition, data in the sample is either edges or subgraphs for the approximate algorithms listed.

The StreamFSM (Ray, Holder & Choudhury, 2014) discovers the frequent subgraphs in a graph, represented by a stream of labeled nodes and edges. In this model, updates to the graph arrive in the form of batches that contain new nodes and edges. The proposed method continuously reports the frequent subgraphs that are estimated to be found in the entire graph as each batch arrives. It is evaluated using five large dynamic graph datasets: the Hetrec 2011 challenge data, Twitter, DBLP and two synthetic datasets. It is evaluated against two popular static large graph miners and experimental results show that it can find the same frequent subgraphs as a non-incremental approach applied to snapshot graphs, and in less time. The drawback of the StreamFSM algorithm: (i) in terms of several parameters that have to be tuned in order to get the optimal performance in terms of time and accuracy/interestingness of results, (ii) it assumes that we have access to the entire graph as the graph grows however this assumption will not work in a real world-streaming scenario, (iii) it only handles increments with additions.

SR and OSR are two dynamic approximate algorithms (Aslay et al., 2018). Both algorithms use reservoir sampling technique (RS) as introduced in Vitter (1985). RS is a fixed size randomized sampling technique; it maintains a fixed-size uniform sample of the data. The sample size is assigned as an input parameter. The algorithms starts and fills the reservoir until it reaches the user defined fixed size. Once the maximum sample size is reached or the reservoir is filled, each new item replaces an existing item in the sample. The addition of an edge affects only the subgraphs in the local neighborhoods up to specified neighborhood. A uniform sample of subgraphs is maintained by iterating through the subgraphs in the neighborhood of the newly inserted edge.

Triest algorithm (De Stefani et al., 2016) counts triangles in incremental streams with fixed memory size, it uses standard reservoir sampling (Vitter, 1985) to maintain the edge sample. The drawback of this algorithm that edge deletion from reservoir is done randomly when reservoir is full, by this random deletion some important patterns might be lost.

TipTap (Nasir et al., 2021) is suite of algorithms designed for mining frequent subgraphs in evolving graphs with high quality approximations. The objective is to maintain a uniform sample of k-vertex subgraphs with a neighborhood exploration procedure. Instead of fixed sample size of earlier algorithms the algorithms in the suit rely on flexible sample size where the bounds determined by Vapnik–Chervonenkis dimension (Riondato & Upfal, 2018). This allows deriving a sufficient sample size that is independent from the size of the graph.