Efficient Enumeration of Maximal Cliques in Large Real-World Link Streams

Alapfogalmak

The algorithm efficiently enumerates all maximal cliques in large real-world link streams by adapting the Bron-Kerbosch algorithm to the dynamic setting and using a novel vertex-maximality test.

Kivonat

The key highlights and insights of the content are:

The authors propose a new algorithm for enumerating maximal cliques in link streams, which is a model for representing interactions over time.

The algorithm first enumerates all time-maximal cliques by processing the instantaneous graphs formed by links existing at each time instant. It then filters these time-maximal cliques to keep only the vertex-maximal ones.

The authors provide a proof of correctness for the algorithm, showing that it lists each maximal clique exactly once.

The complexity analysis shows that the algorithm has better time complexity than the state-of-the-art in many cases, and its output-sensitive complexity is close to the output size, indicating efficiency.

Experimental results demonstrate that the new algorithm significantly outperforms existing methods, allowing enumeration of cliques in link streams that are two orders of magnitude larger than previously feasible.

The authors provide both a Python and a highly efficient C++ implementation of the algorithm, the latter of which also supports parallel enumeration.

Statisztikák

The link stream has the following characteristics:

|T|: number of different time instants at which links begin or end
m = |E|: the number of links
d: the maximal degree of a vertex in any static graph Gt
αT: the number of time-maximal cliques in the link stream
α: the number of maximal cliques (both time-maximal and vertex-maximal) in the link stream
q: the maximal number of vertices in a clique

Idézetek

"Link streams offer a good model for representing interactions over time."
"Listing all maximal cliques in a graph is known to be NP-hard. However, this problem is of utmost importance to analyze real-world graphs because it reveals high-density subgraphs and is thus a keystone to understand their structure."
"We show experimentally that it significantly outperforms the state of the art and allows enumerating cliques in networks that are two orders of magnitude larger than what was previously feasible in the time and memory limits of the protocol."

Főbb Kivonatok

Faster maximal clique enumeration in large real-world link streams

by Alex... : arxiv.org 05-03-2024

https://arxiv.org/pdf/2302.00360.pdf

Mélyebb kérdések

How could the algorithm be extended to handle other types of temporal constraints beyond maximal cliques, such as k-plexes or (∆, γ)-cliques

To extend the algorithm to handle other types of temporal constraints beyond maximal cliques, such as k-plexes or (∆, γ)-cliques, we can modify the vertex maximality test in the algorithm. For k-plexes, we would need to adjust the condition for adding a vertex to a clique to ensure that each vertex interacts with at least s-k vertices in the subgraph. This would involve updating the neighborhood calculation and the final time calculation to consider the additional constraints of k-plexes. Similarly, for (∆, γ)-cliques, we would need to incorporate the requirement that each link appears at least γ times in each sub-interval of size ∆. This would involve adjusting the conditions for adding edges to the clique and checking for the temporal constraints within each sub-interval.

What are the potential applications of the efficient maximal clique enumeration in link streams beyond the analysis of interaction networks

The efficient maximal clique enumeration in link streams has various potential applications beyond the analysis of interaction networks. Some of these applications include:

Community Detection: Maximal cliques can be used as a basis for community detection algorithms in social networks, where densely connected groups of individuals form distinct communities.
Anomaly Detection: Identifying maximal cliques in link streams can help in detecting anomalies or unusual patterns in network interactions, which can be indicative of security breaches or irregular behavior.
Recommendation Systems: Understanding the dense subgraphs represented by maximal cliques can improve recommendation systems by identifying clusters of users with similar preferences or behaviors.
Graph Compression: Maximal cliques can be used to compress large graphs by representing dense subgraphs as single nodes, reducing the complexity of the network while preserving important structural information.
Biological Networks: Analyzing maximal cliques in biological networks can help in understanding protein interactions, gene regulatory networks, and other biological processes.

Can the techniques used in this algorithm be applied to other dynamic graph problems beyond clique enumeration

The techniques used in this algorithm for maximal clique enumeration in link streams can be applied to other dynamic graph problems beyond clique enumeration. Some potential applications include:

Subgraph Matching: Adapting the algorithm to find maximal common subgraphs in evolving graphs, which can be useful in pattern recognition, network alignment, and graph similarity analysis.
Dynamic Community Detection: Extending the algorithm to detect evolving communities in dynamic networks, where communities change over time based on interactions and connections.
Temporal Network Analysis: Applying the techniques to analyze temporal networks for studying information flow, influence propagation, and temporal patterns in network dynamics.
Event Detection: Using the algorithm to identify significant events or changes in network behavior by analyzing the evolution of maximal cliques over time.
Graph Evolution Prediction: Predicting the future evolution of graphs based on the patterns observed in maximal cliques, which can be valuable in forecasting network changes and trends.