On Robust Temporal Structures in Highly Dynamic Networks Arnaud - PowerPoint PPT Presentation

On Robust Temporal Structures in Highly Dynamic Networks Arnaud Casteigts (LaBRI, University of Bordeaux) J. work with Swan Dubois, Franck Petit, and John Michael Robson https://arxiv.org/abs/1703.03190 AATG@ICALP 2018

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Highly dynamic networks Ex: How changes are perceived? - Faults and Failures? - Nature of the system. Change is normal. - Possibly partitioned network Example of scenario

Graph representations Time-varying graphs (TVG) [ 2 , 3 ] [ 0 ] G = ( V , E , T , ρ, ζ ) - T ⊆ N / R (lifetime) [ 1 , 2 ] [ 2 , 3 ] [ 0 , 1 ] [ 0 , 2 ] - ρ : E × T → { 0 , 1 } (presence fonction) - ζ : E × T → N / R (latency function) [ 0 ] Another classical view G = G 0 , G 1 , ... G 0 G 1 G 2 G 3 Variety of models and terminologies: Dynamic graphs, evolving graphs, temporal graphs, link streams, etc. C., Flocchini, Quattrociocchi, Int. J. of Parallel, Emergent and Distributed Systems, Vol. 27, Issue 5, 2012 (among others)

Graph representations Time-varying graphs (TVG) { 1 / i : i ∈ N } [ 5 , 7 ] { t ∈ N : t prime } G = ( V , E , T , ρ, ζ ) - T ⊆ N / R (lifetime) [ 1 , π ] [ 9999 , ∞ ) [ 0 , ∞ ) - ρ : E × T → { 0 , 1 } (presence fonction) - ζ : E × T → N / R (latency function) [ 0 , 1 ] ∪ [ 2 , 5 ] Another classical view G = G 0 , G 1 , ... G 0 G 1 G 2 G 3 Variety of models and terminologies: Dynamic graphs, evolving graphs, temporal graphs, link streams, etc. C., Flocchini, Quattrociocchi, Int. J. of Parallel, Emergent and Distributed Systems, Vol. 27, Issue 5, 2012 (among others)

Graph representations Time-varying graphs (TVG) { 1 / i : i ∈ N } [ 5 , 7 ] { t ∈ N : t prime } G = ( V , E , T , ρ, ζ ) - T ⊆ N / R (lifetime) [ 1 , π ] [ 9999 , ∞ ) [ 0 , ∞ ) - ρ : E × T → { 0 , 1 } (presence fonction) - ζ : E × T → N / R (latency function) [ 0 , 1 ] ∪ [ 2 , 5 ] Another classical view G = G 0 , G 1 , ... G 0 G 1 G 2 G 3 the graph Variety of models and terminologies: Dynamic graphs, evolving graphs, temporal graphs, link streams, etc. C., Flocchini, Quattrociocchi, Int. J. of Parallel, Emergent and Distributed Systems, Vol. 27, Issue 5, 2012 (among others)

Basic concepts a e a e a e a e c c c c b d b d b d b d G 0 G 1 G 2 G 3

Basic concepts a e a e a e a e c c c c b d b d b d b d G 0 G 1 G 2 G 3 = ⇒ Temporal path ( a.k.a. Journey), e.g. a � e Ex: (( ac , t 1 ) , ( cd , t 2 ) , ( de , t 3 )) with t i + 1 ≥ t i and ρ ( e i , t i ) = 1 (can be formulated with latency)

Basic concepts a e a e a e a e c c c c b d b d b d b d G 0 G 1 G 2 G 3 = ⇒ Temporal path ( a.k.a. Journey), e.g. a � e Ex: (( ac , t 1 ) , ( cd , t 2 ) , ( de , t 3 )) with t i + 1 ≥ t i and ρ ( e i , t i ) = 1 (can be formulated with latency) = ⇒ Temporal connectivity ( ∗ � ∗ ) Satisfied here?

Basic concepts a e a e a e a e c c c c b d b d b d b d G 0 G 1 G 2 G 3 = ⇒ Temporal path ( a.k.a. Journey), e.g. a � e Ex: (( ac , t 1 ) , ( cd , t 2 ) , ( de , t 3 )) with t i + 1 ≥ t i and ρ ( e i , t i ) = 1 (can be formulated with latency) = ⇒ Temporal connectivity ( ∗ � ∗ ) Satisfied here? No, only 1 � ∗ .

Basic concepts a e a e a e a e c c c c b d b d b d b d G 0 G 1 G 2 G 3 = ⇒ Temporal path ( a.k.a. Journey), e.g. a � e Ex: (( ac , t 1 ) , ( cd , t 2 ) , ( de , t 3 )) with t i + 1 ≥ t i and ρ ( e i , t i ) = 1 (can be formulated with latency) = ⇒ Temporal connectivity ( ∗ � ∗ ) Satisfied here? No, only 1 � ∗ . = ⇒ Footprint ( � = underlying graph) a e c b d

Today: Covering problems Three ways of redefining covering problems C., Mans, Mathieson, 2011 Ex: D OMINATING S ET G 1 G 2 G 3 Temporal dominating set Evolving dominating set Permanent dominating set

Today: Covering problems Three ways of redefining covering problems C., Mans, Mathieson, 2011 Ex: D OMINATING S ET G 1 G 2 G 3 Temporal dominating set Evolving dominating set Permanent dominating set → How about infinite time? The relation must hold infinitely often!

Classes of dynamic networks (C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem? (based on time-varying graphs)

Classes of dynamic networks (C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem? (C., 2018)

Classes of dynamic networks (C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem? (C., 2018) → E R ≡ all the edges of the footprint are recurrent → T C R ≡ temporal connectivity is recurrently achived

Classes of dynamic networks (C.,Flocchini,Quattrociocchi,Santoro, 2012) What assumption for what problem? (C., 2018) → E R ≡ all the edges of the footprint are recurrent → T C R ≡ temporal connectivity is recurrently achived Building temporal covering structures? → E R : “easy” → T C R : this talk

Exploiting regularities within T C R T C R := Temporal connectivity is recurrently achieved ( T C R := ∀ t , G [ t , + ∞ ) ∈ T C )

Exploiting regularities within T C R T C R := Temporal connectivity is recurrently achieved ( T C R := ∀ t , G [ t , + ∞ ) ∈ T C ) Alternative characterization: T C R ≡ Eventual footprint connected Braud Santoni et al., 2016 a e − → c b d → Can be exploited in a distributed algorithm Kaaouachi et al. , 2016

Exploiting regularities within T C R T C R := Temporal connectivity is recurrently achieved ( T C R := ∀ t , G [ t , + ∞ ) ∈ T C ) Alternative characterization: T C R ≡ Eventual footprint connected Braud Santoni et al., 2016 a e − → c b d → Can be exploited in a distributed algorithm Kaaouachi et al. , 2016 → Robustness: New form of heredity asking that a property or solution holds in all connected spanning subgraphs Ex: M INIMAL D OMINATING S ET (MDS) and M AXIMAL I NDEPENDENT S ET (MIS) C., Dubois, Petit, Robson, 2017/18

E X : M AXIMAL I NDEPENDENT S ETS A maximal independent set (MIS) is a maximal ( � = maximum) set of nodes, none of which are neighbors. (a) (b) (c) (d)

E X : M AXIMAL I NDEPENDENT S ETS A maximal independent set (MIS) is a maximal ( � = maximum) set of nodes, none of which are neighbors. (e) (f) (g) (h) Which ones are robust?

E X : M AXIMAL I NDEPENDENT S ETS A maximal independent set (MIS) is a maximal ( � = maximum) set of nodes, none of which are neighbors. (i) (j) (k) (l) Which ones are robust? → Question: characterizing graphs/footprints in which 1. all MISs are robust: ( RMIS ∀ ) 2. at least one MIS is robust: ( RMIS ∃ ) 3. all MDSs are robust: ( RMDS ∀ ) 4. at least one MDS is robust: ( RMDS ∃ )

Overview of technical results 1. RMDS ∀ = Sputniks 2. RMIS ∀ = Complete bipartite ∪ Sputniks 3. RMDS ∃ � bipartite + test algo 4. RMIS ∃ � bipartite + test algo Locality: RMDS∃ 1. RMDS ∀ and RMIS ∀ RMIS∃ → Robust solutions can be computed locally! RMIS∀ 2. RMIS ∃ → Robust solutions cannot be computed locally! RMDS∀ Local algo for robust MIS in Sputniks Lower bound on the non-locality of robust MIS

RMIS ∀ Graphs in which all MISs are robust? ( RMIS ∀ )

RMIS ∀ Graphs in which all MISs are robust? ( RMIS ∀ ) Lemma Bipartite complete ( BK ) graphs ⊆ RMIS ∀ .

RMIS ∀ Graphs in which all MISs are robust? ( RMIS ∀ ) Lemma Bipartite complete ( BK ) graphs ⊆ RMIS ∀ . Def: A graph is a sputnik if and only if every node that belongs to a cycle also has an antenna ( i.e. a pendant neighbor). Lemma Sputniks ⊆ RMIS ∀ .

RMIS ∀ Graphs in which all MISs are robust? ( RMIS ∀ ) Lemma Bipartite complete ( BK ) graphs ⊆ RMIS ∀ . Def: A graph is a sputnik if and only if every node that belongs to a cycle also has an antenna ( i.e. a pendant neighbor). Lemma Sputniks ⊆ RMIS ∀ . Theorem RMIS ∀ = Sputniks ∪ BK

Local algorithm to find a RMIS in RMIS ∀ State of the art (classical MIS) ◮ Lower bound: Ω( � log n / log log n ) [KMW04] ◮ Best algo: 2 O ( √ log n ) [PS96] (between log n and n ) ◮ Best algo in trees: O (log n / log log n ) [BE10] Can we solve the problem locally in RMIS ∀ ?